Theorem: Let 2 < p [itex] \in \mathbb{P}[/itex] be the odd leg and [itex]\frac{p^2 - 1}{2}[/itex] be the even leg of a rectangular triangle,
then the the hypothenuse will be [itex]\frac{p^2 + 1}{2}[/itex].

Corollary: For '2 primes in PPT' we must have the hypothenuse h = [itex]\frac{p^2 + 1}{2}[/itex] be a prime

The OEIS sequence A048161 ("Primes p such that q=(p^2+1)/2 is also a prime")

The odd leg has to be 1 mod 10 or 9 mod 10. If it is 3 mod 10 or 7 mod 10, the hypotenuse will be a multiple of 5. If it is 5 mod 10 the odd leg isn't prime. There are some small exceptions where the multiple of 5 is exactly 5.

We have a necessary condition, that the odd leg is [itex] \pm 1 [/itex] mod 10, but the
condition is not sufficient, as is shown by the various primes, which are [itex] \pm 1 [/itex] mod 10
but do not fit in the pattern as 31,41,89,109,149,151,179,191 etc